a girl weighing 620 n sits in the middle of a 2.8 m long hammock which sags 1.2m below th e point of support. What force is exerted by each of the two ropes supporting the hammock
To determine the force exerted by each of the two ropes supporting the hammock, we need to calculate the vertical components of tension in the ropes.
First, let's calculate the net force acting on the girl in the vertical direction. We'll assume upward as the positive direction.
The net force (F_net) equals the weight of the girl (620 N) minus the supporting force exerted by the hammock (F_hammock).
F_net = 620 N - F_hammock
Given that the hammock sags 1.2 m below the point of support, the vertical distance between the girl and the point of support is 1.2 m.
The weight of the girl can be calculated using the formula:
Weight = mass × gravitational acceleration
We know the weight (620 N) and the gravitational acceleration (9.8 m/s^2), so we can find the mass of the girl:
620 N = mass × 9.8 m/s^2
mass = 620 N / 9.8 m/s^2
mass = 63.27 kg (approx.)
Now, let's calculate the gravitational force. The gravitational force acting on the girl is equal to her weight (620 N):
Gravitational force = Weight = 620 N
Since the hammock is in equilibrium, the net force in the y-direction must also be zero. Therefore, the upward force exerted by each rope must balance the downward force of gravity.
The hammock forms an isosceles triangle, so the two ropes equally share the load. The angle between each rope and the horizontal is half of the angle at the point of support.
Using trigonometry:
sin(angle/2) = height sag / (length/2)
Now, substituting the given values:
sin(angle/2) = 1.2 m / (2.8 m / 2)
sin(angle/2) = 1.2 m / 1.4 m
sin(angle/2) = 0.8571
To find the angle itself:
angle/2 = arcsin(0.8571)
angle/2 ≈ 58.9°
Now we can calculate the tension in each rope using the angle:
Tension in each rope = Gravitational force / sin(angle)
Tension in each rope = 620 N / sin(58.9°)
Tension in each rope ≈ 715 N
Therefore, the force exerted by each of the two ropes supporting the hammock is approximately 715 N.
To find the force exerted by each of the two ropes supporting the hammock, we need to consider the equilibrium of forces acting on the hammock.
When the girl is sitting in the middle of the hammock, the weight of the girl (620 N) acts vertically downward. This weight force can be broken down into two components:
1. Vertical Component: The vertical component of the weight force is balanced by the tension in the ropes. This component helps to support the weight of the girl without causing any acceleration in the vertical direction.
2. Horizontal Component: The horizontal component of the weight force has no effect on the tension in the ropes, as it acts perpendicular to the direction of the ropes.
Now, let's calculate the force exerted by each of the two ropes:
Step 1: Calculate the vertical component of the weight force.
The vertical component of the weight force can be found using trigonometry. The weight force can be represented as the hypotenuse of a right-angled triangle, with the vertical component as one of the sides and the angle of sag (1.2 m) as the angle between the weight force and the vertical direction.
Using trigonometry, we can calculate the vertical component of the weight force:
Vertical component = Weight force * sin(angle of sag)
Vertical component = 620 N * sin(1.2 m/2.8 m)
Step 2: Calculate the force exerted by each rope.
Since the hammock is in equilibrium, the tension in each rope is equal.
Force exerted by each rope = Vertical component of the weight force / 2
Once you have the value of the vertical component of the weight force (Step 1), you can divide it by 2 to find the force exerted by each rope.